Problem 56 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-81}}{\sqrt{-9}}$$

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Express Each Term Using \(i\)

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). Therefore, you can express \(\sqrt{-81}\) and \(\sqrt{-9}\) in terms of \(i\) by taking the square root of the positive part and multiplying by \(i\).\(\sqrt{-81} = i \sqrt{81} = i imes 9 = 9i\)\(\sqrt{-9} = i \sqrt{9} = i imes 3 = 3i\)

Step 2: Perform the Division

Now divide the expression using the results from Step 1:\(\frac{\sqrt{-81}}{\sqrt{-9}} = \frac{9i}{3i}\)

Step 3: Simplify the Expression

Cancel the \(i\) terms in the numerator and denominator, as long as \(i eq 0\), which simplifies to:\(\frac{9i}{3i} = \frac{9}{3} = 3\)

Step 4: Final Simplified Result

The final simplified result is simply \(3\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the study of complex numbers. It is defined as \(i = \sqrt{-1}\). This definition allows mathematicians to represent the square roots of negative numbers, which are not possible using only real numbers.

When you multiply \(i\) by itself (\(i^2\)), you get \(-1\). This property is very useful for understanding various operations involving complex numbers. Some key points to remember about the imaginary unit \(i\) are:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\) (and the cycle repeats)

When simplifying expressions or performing calculations with complex numbers, knowing these properties of \(i\) can help you understand and solve problems faster.

Simplifying Expressions

Simplifying expressions that involve complex numbers, especially those expressed in terms of \(i\), requires you to handle each component with care. Let's break down how to simplify such expressions, using an example.

To simplify \( \frac{\sqrt{-81}}{\sqrt{-9}} \):

1. Start by expressing each terms using \(i\):
   • \(\sqrt{-81} = i \sqrt{81} = 9i\)
   • \(\sqrt{-9} = i \sqrt{9} = 3i\)
2. Perform the division:
   • \(\frac{9i}{3i}\)
3. Simplify by canceling the \(i\) terms:
   • \(\frac{9}{3} = 3\)

In each step, carefully handle terms to ensure a correct final result. Here, the final simplified answer is \(3\). The idea is to break complex expressions into smaller, more manageable parts, and simplify accordingly.

Square Roots of Negative Numbers

The square roots of negative numbers can initially seem puzzling, but they become simpler when expressed in terms of the imaginary unit \(i\). By definition, the square root of a negative number \(-x\) can be expressed as \(i\sqrt{x}\).

This convention allows you to overcome the limitation of taking the square root of negative numbers in the realm of real numbers. For instance:

• \(\sqrt{-81} = i \sqrt{81} = 9i\)
• \(\sqrt{-4} = i \sqrt{4} = 2i\)

When dealing with square roots of negative numbers, always first separate the negative part using \(i\), then find the square root of the positive part. This approach aids in simplifying complex expressions and is pivotal in solving problems involving complex numbers.